Subgroups — C₃₁×#23024 #21479783

Order 7936.

982 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#467	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃₁×#107	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2360	C₂	1	2	yes	maximal
5	C₃₁×#1969	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1942	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1928	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1872	C₂	1	2	yes	maximal
9	C₃₁×#1863	C₂	1	2	yes	maximal, complemented
10	C₄×C₃₁×#107	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×C₂²⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
15	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
16	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
17	C₃₁×#467	C₄	1	8	yes	complemented
18	C₄×Q₁₆×C₃₁	—	2	4	no	—
19	C₂×C₃₁×#107	—	2	4	no	—
20	C₂×C₃₁×#107	C₂²	1	2	yes	—
21	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#103	C₂²	1	2	yes	—
25	C₂×C₃₁×#103	C₂²	1	2	yes	—
26	C₂×C₃₁×#103	C₂²	1	2	yes	—
27	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
30	C₃₁×#391	C₂²	1	2	yes	—
31	C₃₁×#389	—	2	2	no	—
32	C₃₁×#389	C₂²	1	2	yes	—
33	C₃₁×#387	—	2	2	no	—
34	C₃₁×#385	—	2	2	no	—
35	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
36	C₄×C₂²⋊C₄×C₃₁	—	2	2	no	—
37	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
38	C₃₁×#340	C₂²	1	2	yes	—
39	C₃₁×#340	C₂²	1	2	yes	—
40	C₃₁×#336	C₂²	1	1	yes	characteristic
41	C₃₁×#336	C₂²	1	1	yes	characteristic
42	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
43	C₂³×C₄×C₃₁	—	2	2	no	—
44	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
45	C₂×Q₁₆×C₃₁	—	2	2	no	—
46	C₂×Q₁₆×C₃₁	—	2	2	no	—
47	C₂×Q₁₆×C₃₁	C₂×C₄	1	4	yes	complemented
48	C₂²×C₈×C₃₁	—	2	2	no	—
49	C₂²×C₈×C₃₁	—	2	2	no	—
50	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
51	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
55	C₃₁×C₂²⋊Q₈	—	2	4	no	—
56	C₃₁×C₂²⋊Q₈	C₂×C₄	1	8	yes	—
57	C₄×Q₈×C₃₁	—	2	4	no	—
58	C₄×Q₈×C₃₁	—	4	2	no	—
59	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
60	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
61	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
62	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
63	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
64	C₂×C₄⋊C₄×C₃₁	—	4	2	no	—
65	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
68	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
69	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
70	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
71	C₂×C₄²×C₃₁	—	2	2	no	—
72	C₂×C₄²×C₃₁	—	2	2	no	—
73	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
74	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
75	C₃₁×#107	—	2	2	no	—
76	C₃₁×#107	—	2	4	no	—
77	C₃₁×#107	C₂×C₄	1	4	yes	complemented
78	C₃₁×#107	D₄	1	4	yes	—
79	C₃₁×#103	—	2	4	no	—
80	C₃₁×#103	—	4	2	no	—
81	C₃₁×#103	C₂×C₄	1	8	yes	complemented
82	C₃₁×C₂²⋊C₈	—	2	2	no	—
83	C₄×C₈×C₃₁	—	2	2	no	—
84	C₄×C₈×C₃₁	—	2	2	no	—
85	C₃₁×#95	—	2	2	no	—
86	C₃₁×#95	—	2	2	no	—
87	C₃₁×#95	—	2	2	no	—
88	C₃₁×#95	—	2	2	no	—
89	C₃₁×#95	D₄	1	1	yes	characteristic
90	C₃₁×#95	D₄	1	1	yes	characteristic
91	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
92	C₂×Q₈×C₃₁	—	2	2	no	—
93	C₂×Q₈×C₃₁	—	2	2	no	—
94	C₂×Q₈×C₃₁	—	2	2	no	—
95	C₂×Q₈×C₃₁	—	2	2	no	—
96	C₂×Q₈×C₃₁	—	2	2	no	—
97	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
98	C₂²×C₄×C₃₁	—	2	1	no	—
99	C₂²×C₄×C₃₁	—	2	1	no	—
100	C₂²×C₄×C₃₁	—	2	1	no	—
101	C₂²×C₄×C₃₁	—	2	1	no	—
102	C₂²×C₄×C₃₁	—	2	2	no	—
103	C₂²×C₄×C₃₁	—	2	2	no	—
104	C₂²×C₄×C₃₁	—	2	2	no	—
105	C₂²×C₄×C₃₁	—	2	2	no	—
106	C₂²×C₄×C₃₁	—	4	4	no	—
107	C₂²×C₄×C₃₁	—	4	4	no	—
108	C₂²×C₄×C₃₁	—	8	2	no	—
109	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
110	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₃₁	#41	1	2	yes	—
116	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
117	Q₁₆×C₃₁	—	2	8	no	—
118	C₂×C₈×C₃₁	—	2	1	no	—
119	C₂×C₈×C₃₁	—	2	1	no	—
120	C₂×C₈×C₃₁	—	2	8	no	—
121	C₂×C₈×C₃₁	—	2	1	no	—
122	C₂×C₈×C₃₁	—	2	1	no	—
123	C₂×C₈×C₃₁	—	2	1	no	—
124	C₂×C₈×C₃₁	—	2	1	no	—
125	C₂×C₈×C₃₁	—	2	1	no	—
126	C₂×C₈×C₃₁	—	2	1	no	—
127	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
128	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
129	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
130	C₂×C₈×C₃₁	#41	1	2	yes	—
131	C₄⋊C₄×C₃₁	—	2	8	no	—
132	C₄⋊C₄×C₃₁	—	2	2	no	—
133	C₄⋊C₄×C₃₁	—	2	2	no	—
134	C₄⋊C₄×C₃₁	—	2	2	no	—
135	C₄⋊C₄×C₃₁	—	2	2	no	—
136	C₄⋊C₄×C₃₁	—	4	4	no	—
137	C₄⋊C₄×C₃₁	—	4	4	no	—
138	C₄⋊C₄×C₃₁	—	8	2	no	—
139	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
140	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
141	C₂²⋊C₄×C₃₁	—	2	8	no	—
142	C₂²⋊C₄×C₃₁	—	2	2	no	—
143	C₂²⋊C₄×C₃₁	Q₁₆	1	4	yes	complemented
144	C₄²×C₃₁	—	2	1	no	—
145	C₄²×C₃₁	—	2	1	no	—
146	C₄²×C₃₁	—	2	1	no	—
147	C₄²×C₃₁	—	2	1	no	—
148	C₄²×C₃₁	—	4	4	no	—
149	C₄²×C₃₁	—	8	2	no	—
150	#23024	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₂³×C₃₁	—	2	1	no	—
152	C₂³×C₃₁	—	2	1	no	—
153	C₂³×C₃₁	—	2	1	no	—
154	C₂³×C₃₁	—	2	1	no	—
155	C₂³×C₃₁	C₄×D₄	1	2	yes	—
156	C₂³×C₃₁	C₂×Q₁₆	1	2	yes	—
157	C₂³×C₃₁	#135	1	2	yes	—
158	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
159	Q₈×C₃₁	—	2	4	no	—
160	Q₈×C₃₁	—	2	4	no	—
161	Q₈×C₃₁	—	4	2	no	—
162	Q₈×C₃₁	—	4	2	no	—
163	C₂×C₄×C₃₁	—	2	1	no	—
164	C₂×C₄×C₃₁	—	2	1	no	—
165	C₂×C₄×C₃₁	—	2	1	no	—
166	C₂×C₄×C₃₁	—	2	1	no	—
167	C₂×C₄×C₃₁	—	2	1	no	—
168	C₂×C₄×C₃₁	—	2	1	no	—
169	C₂×C₄×C₃₁	—	2	2	no	—
170	C₂×C₄×C₃₁	—	2	2	no	—
171	C₂×C₄×C₃₁	—	2	2	no	—
172	C₂×C₄×C₃₁	—	2	2	no	—
173	C₂×C₄×C₃₁	—	2	4	no	—
174	C₂×C₄×C₃₁	—	2	4	no	—
175	C₂×C₄×C₃₁	—	2	1	no	—
176	C₂×C₄×C₃₁	—	2	4	no	—
177	C₂×C₄×C₃₁	—	2	1	no	—
178	C₂×C₄×C₃₁	—	2	1	no	—
179	C₂×C₄×C₃₁	—	2	1	no	—
180	C₂×C₄×C₃₁	—	2	1	no	—
181	C₂×C₄×C₃₁	—	2	1	no	—
182	C₂×C₄×C₃₁	—	4	2	no	—
183	C₂×C₄×C₃₁	—	4	2	no	—
184	C₂×C₄×C₃₁	—	4	2	no	—
185	C₂×C₄×C₃₁	—	4	2	no	—
186	C₂×C₄×C₃₁	—	4	2	no	—
187	C₂×C₄×C₃₁	—	4	2	no	—
188	C₂×C₄×C₃₁	—	4	2	no	—
189	C₂×C₄×C₃₁	—	4	2	no	—
190	C₂×C₄×C₃₁	—	4	2	no	—
191	C₂×C₄×C₃₁	—	8	8	no	—
192	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
193	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
194	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
195	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
196	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
197	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
198	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
199	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
200	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
201	C₈×C₃₁	—	2	2	no	—
202	C₈×C₃₁	—	4	1	no	—
203	C₈×C₃₁	—	4	1	no	—
204	C₈×C₃₁	C₄×D₄	1	4	yes	—
205	C₂×#467	C₂×C₃₁	1	1	yes	characteristic
206	C₂×C₄×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
207	C₂²×#107	C₂×C₃₁	1	1	yes	characteristic
208	#2360	C₂×C₃₁	1	2	yes	—
209	#1969	C₂×C₃₁	1	1	yes	characteristic, complemented
210	#1942	C₂×C₃₁	1	1	yes	characteristic, complemented
211	#1928	C₂×C₃₁	1	2	yes	complemented
212	#1872	C₂×C₃₁	1	2	yes	—
213	#1863	C₂×C₃₁	1	2	yes	complemented
214	C₄×#107	C₂×C₃₁	1	1	yes	characteristic, complemented
215	C₈×C₂²⋊C₄	C₂×C₃₁	1	1	yes	characteristic
216	C₂²×C₃₁	—	2	2	no	—
217	C₂²×C₃₁	—	2	2	no	—
218	C₂²×C₃₁	—	2	2	no	—
219	C₂²×C₃₁	—	2	2	no	—
220	C₂²×C₃₁	—	2	2	no	—
221	C₂²×C₃₁	—	2	2	no	—
222	C₂²×C₃₁	C₄×Q₁₆	1	4	yes	complemented
223	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
224	C₂²×C₃₁	D₄²	1	1	yes	characteristic
225	C₂²×C₃₁	#547	1	1	yes	characteristic
226	C₂²×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
227	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
228	C₂²×C₃₁	#576	1	1	yes	characteristic
229	C₂²×C₃₁	#578	1	1	yes	characteristic
230	C₄×C₃₁	—	2	2	no	—
231	C₄×C₃₁	—	2	2	no	—
232	C₄×C₃₁	—	2	2	no	—
233	C₄×C₃₁	—	4	1	no	—
234	C₄×C₃₁	—	4	4	no	—
235	C₄×C₃₁	—	4	4	no	—
236	C₄×C₃₁	—	4	1	no	—
237	C₄×C₃₁	—	4	4	no	—
238	C₄×C₃₁	—	8	2	no	—
239	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
240	C₄×C₃₁	#520	1	1	yes	characteristic
241	C₄×C₃₁	#548	1	1	yes	characteristic
242	C₄×C₃₁	#553	1	1	yes	characteristic
243	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
244	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
245	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
246	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
247	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
248	#467	C₄×C₃₁	1	8	yes	complemented
249	C₄×Q₁₆	—	2	4	no	—
250	C₂×#107	—	2	4	no	—
251	C₂×#107	C₂²×C₃₁	1	2	yes	—
252	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
253	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
254	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
255	C₂×#103	C₂²×C₃₁	1	2	yes	—
256	C₂×#103	C₂²×C₃₁	1	2	yes	—
257	C₂×#103	C₂²×C₃₁	1	2	yes	—
258	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
259	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
260	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
261	#391	C₂²×C₃₁	1	2	yes	—
262	#389	—	2	2	no	—
263	#389	C₂²×C₃₁	1	2	yes	—
264	#387	—	2	2	no	—
265	#385	—	2	2	no	—
266	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
267	C₄×C₂²⋊C₄	—	2	2	no	—
268	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
269	#340	C₂²×C₃₁	1	2	yes	—
270	#340	C₂²×C₃₁	1	2	yes	—
271	#336	C₂²×C₃₁	1	1	yes	characteristic
272	#336	C₂²×C₃₁	1	1	yes	characteristic
273	C₂×C₃₁	—	2	4	no	—
274	C₂×C₃₁	#2361	1	1	yes	characteristic
275	C₂×C₃₁	C₂×C₄×Q₁₆	1	1	yes	characteristic
276	C₂×C₃₁	#2951	1	1	yes	characteristic
277	C₂×C₃₁	#3280	1	1	yes	characteristic
278	C₂×C₃₁	D₄×Q₁₆	1	1	yes	characteristic
279	C₂×C₃₁	#3336	1	1	yes	characteristic
280	C₂×C₃₁	#3338	1	1	yes	characteristic
281	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
282	C₂³×C₄	—	2	2	no	—
283	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
284	C₂×Q₁₆	—	2	2	no	—
285	C₂×Q₁₆	—	2	2	no	—
286	C₂×Q₁₆	C₂×C₄×C₃₁	1	4	yes	complemented
287	C₂²×C₈	—	2	2	no	—
288	C₂²×C₈	—	2	2	no	—
289	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
290	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
291	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
292	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
293	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
294	C₂²⋊Q₈	—	2	4	no	—
295	C₂²⋊Q₈	C₂×C₄×C₃₁	1	8	yes	—
296	C₄×Q₈	—	2	4	no	—
297	C₄×Q₈	—	4	2	no	—
298	C₂×C₄⋊C₄	—	2	2	no	—
299	C₂×C₄⋊C₄	—	2	4	no	—
300	C₂×C₄⋊C₄	—	2	2	no	—
301	C₂×C₄⋊C₄	—	2	2	no	—
302	C₂×C₄⋊C₄	—	2	2	no	—
303	C₂×C₄⋊C₄	—	4	2	no	—
304	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
305	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
306	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
307	C₂×C₂²⋊C₄	—	2	2	no	—
308	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
309	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
310	C₂×C₄²	—	2	2	no	—
311	C₂×C₄²	—	2	2	no	—
312	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
313	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
314	#107	—	2	2	no	—
315	#107	—	2	4	no	—
316	#107	C₂×C₄×C₃₁	1	4	yes	complemented
317	#107	D₄×C₃₁	1	4	yes	—
318	#103	—	2	4	no	—
319	#103	—	4	2	no	—
320	#103	C₂×C₄×C₃₁	1	8	yes	complemented
321	C₂²⋊C₈	—	2	2	no	—
322	C₄×C₈	—	2	2	no	—
323	C₄×C₈	—	2	2	no	—
324	#95	—	2	2	no	—
325	#95	—	2	2	no	—
326	#95	—	2	2	no	—
327	#95	—	2	2	no	—
328	#95	D₄×C₃₁	1	1	yes	characteristic
329	#95	D₄×C₃₁	1	1	yes	characteristic
330	C₃₁	#23024	1	1	yes	characteristic, direct product factor, Sylow
331	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
332	C₂×Q₈	—	2	2	no	—
333	C₂×Q₈	—	2	2	no	—
334	C₂×Q₈	—	2	2	no	—
335	C₂×Q₈	—	2	2	no	—
336	C₂×Q₈	—	2	2	no	—
337	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
338	C₂²×C₄	—	2	1	no	—
339	C₂²×C₄	—	2	1	no	—
340	C₂²×C₄	—	2	1	no	—
341	C₂²×C₄	—	2	1	no	—
342	C₂²×C₄	—	2	2	no	—
343	C₂²×C₄	—	2	2	no	—
344	C₂²×C₄	—	2	2	no	—
345	C₂²×C₄	—	2	2	no	—
346	C₂²×C₄	—	4	4	no	—
347	C₂²×C₄	—	4	4	no	—
348	C₂²×C₄	—	8	2	no	—
349	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
350	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
351	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
352	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
353	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
354	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
355	C₂²×C₄	#41×C₃₁	1	2	yes	—
356	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
357	Q₁₆	—	2	8	no	—
358	C₂×C₈	—	2	1	no	—
359	C₂×C₈	—	2	1	no	—
360	C₂×C₈	—	2	8	no	—
361	C₂×C₈	—	2	1	no	—
362	C₂×C₈	—	2	1	no	—
363	C₂×C₈	—	2	1	no	—
364	C₂×C₈	—	2	1	no	—
365	C₂×C₈	—	2	1	no	—
366	C₂×C₈	—	2	1	no	—
367	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
368	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
369	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
370	C₂×C₈	#41×C₃₁	1	2	yes	—
371	C₄⋊C₄	—	2	8	no	—
372	C₄⋊C₄	—	2	2	no	—
373	C₄⋊C₄	—	2	2	no	—
374	C₄⋊C₄	—	2	2	no	—
375	C₄⋊C₄	—	2	2	no	—
376	C₄⋊C₄	—	4	4	no	—
377	C₄⋊C₄	—	4	4	no	—
378	C₄⋊C₄	—	8	2	no	—
379	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
380	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
381	C₂²⋊C₄	—	2	8	no	—
382	C₂²⋊C₄	—	2	2	no	—
383	C₂²⋊C₄	Q₁₆×C₃₁	1	4	yes	complemented
384	C₄²	—	2	1	no	—
385	C₄²	—	2	1	no	—
386	C₄²	—	2	1	no	—
387	C₄²	—	2	1	no	—
388	C₄²	—	4	4	no	—
389	C₄²	—	8	2	no	—
390	C₂³	—	2	1	no	—
391	C₂³	—	2	1	no	—
392	C₂³	—	2	1	no	—
393	C₂³	—	2	1	no	—
394	C₂³	C₄×D₄×C₃₁	1	2	yes	—
395	C₂³	C₂×Q₁₆×C₃₁	1	2	yes	—
396	C₂³	C₃₁×#135	1	2	yes	—
397	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
398	Q₈	—	2	4	no	—
399	Q₈	—	2	4	no	—
400	Q₈	—	4	2	no	—
401	Q₈	—	4	2	no	—
402	C₂×C₄	—	2	1	no	—
403	C₂×C₄	—	2	1	no	—
404	C₂×C₄	—	2	1	no	—
405	C₂×C₄	—	2	1	no	—
406	C₂×C₄	—	2	1	no	—
407	C₂×C₄	—	2	1	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	2	4	no	—
413	C₂×C₄	—	2	4	no	—
414	C₂×C₄	—	2	1	no	—
415	C₂×C₄	—	2	4	no	—
416	C₂×C₄	—	2	1	no	—
417	C₂×C₄	—	2	1	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	4	2	no	—
422	C₂×C₄	—	4	2	no	—
423	C₂×C₄	—	4	2	no	—
424	C₂×C₄	—	4	2	no	—
425	C₂×C₄	—	4	2	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	4	2	no	—
428	C₂×C₄	—	4	2	no	—
429	C₂×C₄	—	4	2	no	—
430	C₂×C₄	—	8	8	no	—
431	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
432	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
433	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
434	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
435	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
436	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
437	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
438	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
439	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
440	C₈	—	2	2	no	—
441	C₈	—	4	1	no	—
442	C₈	—	4	1	no	—
443	C₈	C₄×D₄×C₃₁	1	4	yes	—
444	C₂²	—	2	2	no	—
445	C₂²	—	2	2	no	—
446	C₂²	—	2	2	no	—
447	C₂²	—	2	2	no	—
448	C₂²	—	2	2	no	—
449	C₂²	—	2	2	no	—
450	C₂²	C₄×Q₁₆×C₃₁	1	4	yes	complemented
451	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
452	C₂²	D₄²×C₃₁	1	1	yes	characteristic
453	C₂²	C₃₁×#547	1	1	yes	characteristic
454	C₂²	C₂²×Q₁₆×C₃₁	1	1	yes	characteristic
455	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
456	C₂²	C₃₁×#576	1	1	yes	characteristic
457	C₂²	C₃₁×#578	1	1	yes	characteristic
458	C₄	—	2	2	no	—
459	C₄	—	2	2	no	—
460	C₄	—	2	2	no	—
461	C₄	—	4	1	no	—
462	C₄	—	4	4	no	—
463	C₄	—	4	4	no	—
464	C₄	—	4	1	no	—
465	C₄	—	4	4	no	—
466	C₄	—	8	2	no	—
467	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
468	C₄	C₃₁×#520	1	1	yes	characteristic
469	C₄	C₃₁×#548	1	1	yes	characteristic
470	C₄	C₃₁×#553	1	1	yes	characteristic
471	C₂	—	2	4	no	—
472	C₂	C₃₁×#2361	1	1	yes	characteristic
473	C₂	C₂×C₄×Q₁₆×C₃₁	1	1	yes	characteristic
474	C₂	C₃₁×#2951	1	1	yes	characteristic
475	C₂	C₃₁×#3280	1	1	yes	characteristic
476	C₂	D₄×Q₁₆×C₃₁	1	1	yes	characteristic
477	C₂	C₃₁×#3336	1	1	yes	characteristic
478	C₂	C₃₁×#3338	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄×C₂²⋊C₄×C₃₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×C₄×Q₁₆×C₃₁	C₂	semidirect
5. C₃₁×#1969	C₂	semidirect
6. C₃₁×#1942	C₂	semidirect
7. C₃₁×#1928	C₂	semidirect
9. C₃₁×#1863	C₂	semidirect
10. C₄×C₃₁×#107	C₂	semidirect
17. C₃₁×#467	C₄	semidirect
47. C₂×Q₁₆×C₃₁	C₂×C₄	semidirect
77. C₃₁×#107	C₂×C₄	semidirect
81. C₃₁×#103	C₂×C₄	semidirect
143. C₂²⋊C₄×C₃₁	Q₁₆	semidirect
150. #23024	C₃₁	direct (N × H)
206. C₂×C₄×Q₁₆	C₂×C₃₁	semidirect
209. #1969	C₂×C₃₁	semidirect
210. #1942	C₂×C₃₁	semidirect
211. #1928	C₂×C₃₁	semidirect
213. #1863	C₂×C₃₁	semidirect
214. C₄×#107	C₂×C₃₁	semidirect
222. C₂²×C₃₁	C₄×Q₁₆	semidirect
248. #467	C₄×C₃₁	semidirect
286. C₂×Q₁₆	C₂×C₄×C₃₁	semidirect
316. #107	C₂×C₄×C₃₁	semidirect
320. #103	C₂×C₄×C₃₁	semidirect
330. C₃₁	#23024	direct (N × H)
383. C₂²⋊C₄	Q₁₆×C₃₁	semidirect
450. C₂²	C₄×Q₁₆×C₃₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃₁×#467	2
C₂×C₄×Q₁₆×C₃₁	2
C₂²×C₃₁×#107	2
C₃₁×#2360	2
C₃₁×#1969	2
C₃₁×#1942	2
C₃₁×#1928	2
C₃₁×#1872	2
C₃₁×#1863	2
C₄×C₃₁×#107	2
C₈×C₂²⋊C₄×C₃₁	2
#23024	31

Subgroups — C31×#23024 #21479783

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23024 #21479783